Thé (sò͘-ha̍k)

Thé sī sò͘-ha̍k lāi-tóe hō͘-chò thiu-siōng tāi-sò͘ ê chi̍t-ki teh gián-kiù ê mi̍h-kiaⁿ, chi̍t-ê chi̍p-ha̍p lāi-tóe ê sêng-ôan nā-sī ū ka-hoat, kiám-hoat, sêng-hoat kap tû-hoat, chia-ê ūn-sǹg koh kap sǹg-sò͘ ê ūn-sǹg sio-siâng-á sio-siâng, án-ne chit-ê chi̍p-ha̍p tio̍h sī thé.

Sò͘-ha̍k ê lē sī ēng ${\displaystyle \mathbb {K} }$ lâi piáu-sī thé, che sī tùi Tek-gí ê Körper lâi-·ê, ì-sù sī 'sin-khu'.

Thé sī chi̍t-ê ū sêng-hoat tan-ūi-ôan ê kau-ōaⁿ-khôan (F, +, ×), jî-chhiáⁿ

• sêng-hoat tan-ūi-ôan 1 kap ka-hoat tan-ūi-ôan 0 sī bô-kâng ê mi̍h-kiāⁿ,
• chún-kóng a∈F, a m̄-sī ka-hoat-tan-ūi-ôan, lán tiāⁿ-tio̍h ōe-tàng tī F nih chhōe-tio̍h chi̍t-ê b, hō͘ a×b=1, ia̍h-tio̍h-sī kóng a ū sêng-hoat ge̍k-ôan.

Chiàu thé ê tēng-gī, F ê jīm-hô 1-ê seng-oân a kap m̄-sī ka-hoat tan-ūi-oân ê b ōe-tàng the̍h-lâi tû, tû liáu-āu ōe tit-tio̍h 1-ê siong. Siong ê tēng-gī sī

${\displaystyle {\frac {a}{b}}=}$b^-1×a

Chit-ê b^-1 sī b ê sêng-hoat ge̍k-oân.

Tû-hoat bô kau-oāⁿ hoat-chek mā bô kiat-ha̍p hoat-chek. I tùi ka-hoat ū hiòng chiàⁿ-chhiú-pêng ê hun-phòe hoat-chek, tio̍h-sī

${\displaystyle {\frac {(a+b)}{c}}={\frac {a}{c}}+{\frac {b}{c}}}$

1. Ho̍k-cha̍p-sò͘ ${\displaystyle \mathbb {C} }$ kap it-poaⁿ teh ēng ê ka-hoat kap sêng-hoat. I koh ū kúi-nā-ê pō͘-hūn-thé, chhin-chhiūⁿ iú-lí-sò͘ ${\displaystyle \mathbb {Q} }$ kap sı̍t-sò͘ ${\displaystyle \mathbb {R} }$.
2. Galois thé sī ū chīn-pōng ê thé, tī sò͘-lūn, tiān-naú-kho-ha̍k kap chu-sìn kang-thêng sī chin tiōng-iàu ê mi̍h-kiāⁿ.

• Chi̍t-ê thé F lāi-tóe m̄-sī 0 ê sêng-ôan chiâⁿ-sêng ê chi̍p-ha̍p ( it-poaⁿ sī kì-chò F^×), kap F ê sêng-hoat chiâⁿ-chò chi̍t-ê abelian kûn. F^× ê pō͘-hūn-kûn sī sûn-khôan-kûn.
• Thé ê piau-sò͘ nā m̄-sī 0 tio̍h-sī sò͘-sò͘.
• Iú-hān-thé ê seng-ôan sò͘-ba̍k it-tēng sī sò͘-sò͘ ê chéng-sò͘ chhù-hong.
• Kā thé tòng-chò khôan, i kan-taⁿ u 2-ê lí-sióng, {0} kap i ka-tī.